Single and multi-specimenR-curve methods forJ IC determination of toughened nylons
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From a fracture mechanical point of view, the fracture propagation in rock material and hence how easily rock can fracture is described by fracture toughness. Fracture toughness is a parameter that describes the resistance of the rock to the propagation of a fracture. In the theory of fracture mechanics, the stress intensity factor K is a measure of the amount of stress concentration at the tip of a crack as a function of applied load and fracture length. The fracture toughness KC is the critical value of the stress intensity factor at which an existing fracture extends. From this mathematical framework it derives that longer fractures are in general easier to propagate.
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Strength of materials
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Orthotropic material
Strain energy
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1. Introduction.- 1.1. Fracture control.- 1.2. The two objectives of damage tolerance analysis.- 1.3. Crack growth and fracture.- 1.4. Damage tolerance and fracture mechanics.- 1.5. The need for analysis: purpose of this book.- 1.6. Exercises.- 2. Effects of Cracks and Notches: Collapse.- 2.1. Scope.- 2.2. An interrupted load path.- 2.3. Stress concentration factor.- 2.4. State of stress at a stress concentration.- 2.5. Yielding at a notch.- 2.6. Plastic collapse at a notch.- 2.7. Fracture at notches: brittle behavior.- 2.8. Measurement of collapse strength.- 2.9. Exercises.- 3. Linear Elastic Fracture Mechanics.- 3.1. Scope.- 3.2. Stress at a crack tip.- 3.3. General form of the stress intensity factor.- 3.4. Toughness.- 3.5. Plastic zone and stresses in plane stress and plane strain.- 3.6. Thickness dependence of toughness.- 3.7. Measurement of toughness.- 3.8. Competition with plastic collapse.- 3.9. The energy criterion.- 3.10. The energy release rate.- 3.11. The meaning of the energy criterion.- 3.12. The rise in fracture resistance: redefinition of toughness.- 3.13. Exercises.- 4. Elastic-Plastic Fracture Mechanics.- 4.1. Scope.- 4.2. The energy criterion for plastic fracture.- 4.3. The fracture criterion.- 4.4. The rising fracture energy.- 4.5. The residual strength diagram in EPFM: collapse.- 4.6. The measurement of the toughness in EPFM.- 4.7. The parameters of the stress-strain curve.- 4.8. The h-functions.- 4.9. Accuracy.- 4.10. Historical development of J.- 4.11. Limitations of EPFM.- 4.12. CTOD measurements.- 4.13. Exercises.- 5. Crack Growth Analysis Concepts.- 5.1. Scope.- 5.2. The concept underlying fatigue crack growth.- 5.3. Measurement of the rate function.- 5.4. Rate equations.- 5.5. Constant amplitude crack growth in a structure.- 5.6. Load interaction: Retardation.- 5.7. Retardation models.- 5.8. Crack growth analysis for variable amplitude loading.- 5.9. Parameters affecting fatigue crack growth rates.- 5.10. Stress corrosion cracking.- 5.11. Exercises.- 6. Load Spectra and Stress Histories.- 6.1. Scope.- 6.2. Types of stress histories.- 6.3. Obtaining load spectra.- 6.4. Exceedance diagram.- 6.5. Stress history generation.- 6.6. Clipping.- 6.7. Truncation.- 6.8. Manipulation of stress history.- 6.9. Environmental effects.- 6.10. Standard spectra.- 6.11. Exercises.- 7. Data Interpretation and Use.- 7.1. Scope.- 7.2. Plane strain fracture toughness.- 7.3. Plane stress and transitional toughness, R-curve.- 7.4. Toughness in terms of J and JR.- 7.5. Estimates of toughness.- 7.6. General remarks on fatigue rate data.- 7.7. Fitting the da/dN data.- 7.8. Dealing with scatter in rate data.- 7.9. Accounting for the environmental effect.- 7.10. Obtaining retardation parameters.- 7.11. Exercises.- 8. Geometry Factors.- 8.1. Scope.- 8.2. The reference stress.- 8.3. Compounding.- 8.4. Superposition.- 8.5. A simple method for asymmetric loading cases.- 8.6. Some easy guesses.- 8.7. Simple solutions for holes and stress concentrations.- 8.8. Simple solutions for irregular stress distributions.- 8.9. Finite element analysis.- 8.10. Simple solutions for crack arresters and multiple elements.- 8.11. Geometry factors for elastic-plastic fracture mechanics.- 8.12. Exercises.- 9. Special Subjects.- 9.1. Scope.- 9.2. Behavior of surface flaws and corner cracks.- 9.3. Break through: leak-before-break.- 9.4. Fracture arrest.- 9.5. Multiple elements, multiple cracks, changing geometry.- 9.6. Stop holes, cold worked holes and interference fasteners.- 9.7. Residual stresses in general.- 9.8. Other loading modes: mixed mode loading.- 9.9. Composites.- 9.10. Exercises.- 10. Analysis Procedures.- 10.1. Scope.- 10.2. Ingredients and critical locations.- 10.3. Critical locations and flaw assumptions.- 10.4. LEFM versus EPFM.- 10.5. Residual strength analysis.- 10.6. Use of R-curve and JR-curve.- 10.7. Crack growth analysis.- 10.8. Exercises.- 11. Fracture Control.- 11.1. Scope.- 11.2. Fracture control options.- 11.3. The probability of missing the crack.- 11.4. The physics and statistics of crack detection.- 11.5. Determining the inspection interval.- 11.6. Fracture control plans.- 11.7. Repairs.- 11.8. Statistical aspects.- 11.9. The cost of fracture and fracture control.- 11.10. Exercises.- 12. Damage Tolerance Substantiation.- 12.1. Scope.- 12.2. Objectives.- 12.3. Analysis and damage tolerance substantiation.- 12.4. Options to improve damage tolerance.- 12.5. Aircraft damage tolerance requirements.- 12.6. Other requirements.- 12.7. Flaw assumptions.- 12.8. Sources of error and safety factors.- 12.9. Misconceptions.- 12.10. Outlook.- 12.11. Exercises.- 13. After the Fact: Fracture Mechanics and Failure Analysis.- 13.1. Scope.- 13.2. The cause of service fractures.- 13.3. Fractography.- 13.4. Features of use in fracture mechanics analysis.- 13.5. Use of fracture mechanics.- 13.6. Possible actions based on failure analysis.- 13.7. Exercises.- 14. Applications.- 14.1. Scope.- 14.2. Storage tank (fictitious example).- 14.3. Fracture arrest in ships.- 14.4. Piping in chemical plant (fictitious example).- 14.5. Fatigue cracks in railroad rails.- 14.6. Underwater pipeline.- 14.7. Closure.- 15. Solutions To Exercises.
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A recently proposed method is used to measure dynamic fracture parameters of Stanstead granite(SG), such as fracture initiation toughness,fracture energy,fracture propagation toughness,and fracture propagation velocity. With the method,a notched semicircular bend specimen is loaded by the split Hopkinson pressure bar(SHPB) system;and a laser gap gauge system is employed to monitor the crack surface opening displacement (CSOD) of the specimen. The dynamic initiation toughness is subsequently calculated by using a quasi-static formula,for the dynamic force balance is achieved in the test. Based on the CSOD data,residual kinetic energies of the two fragments are estimated. The average propagation fracture energy and thus the propagation toughness are calculated. The average fracture propagation velocity is estimated with a series of crack gauges glued on the specimen. It is shown from the results of the experiments that both the initiation and propagation toughnesses of this brittle solid are loadingdependent. The propagation toughness is larger than the initiation toughness. The propagation fracture toughness is shown to increase with the fracture propagation velocity. The fracture arrest toughness and the limiting fracture propagation velocity are obtained by a literature model of this relationship. The results of SG in this work are compared with the ones of Laurentian granite(LG). The grain size of SG is much larger than the one of LG,so that the fracture is easy to be generated and arrested but difficult to propagate. On the contract,the LG has smaller dynamic propagation toughness and the larger limit fracture propagation velocity.
Split Hopkinson Pressure Bar
Brittleness
Compact tension specimen
Charpy impact test
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Abstract This paper contains an assessment of the importance of fracture toughness in hydraulic fracturing. The results given in this paper show that fracture toughness can significantly affect both the height growth and the propagation of hydraulic fractures. Most researchers have considered the effect of fracture toughness on the geometry of the fracture to be negligible except in the cases of small fractures, during injection of very low viscosity fluids, or during fracture closure1. In addition, energy considerations for twodimensional fractures indicated that the energy losses associated with fluid flow were one to two orders of magnitude higher than the energy used in creating new fracture surfaces,γF.2,3 Because γF is directly related to fracture toughness, the conclusion that fracture toughness has a negligible effect on fracture propagation was reaffirmed. However, neglecting the effects of fracture toughness in hydraulic fracturing needs to be reconsidered, because the magnitudes used in earlier calculations may not be representative of in-situ values, and because the development of fully threedimensional models makes possible a more complete study of the problem. The definition of fracture toughness is obtained from the concept of the stress intensity factor, developed in linear elastic fracture mechanics. Preexisting defect are assumed to exist and to induce high stress concentrations in their vicinity, becoming sites for crack initiation and propagation. A single such defect may be represented by a sharp line crack (known as a Griffith crack) in a linear elastic medium. Irwin4 demonstrated that, for a linear elastic homogeneous material, the stresses in the vicinity of a Griffith crack tip vary as 1/r, where r is measured from the crack tip. For a fracture in opening mode (or mode I, see Figure 1), the stresses are given by:
Linear elasticity
Well stimulation
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